| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
|
| Initial Amendment Date: | March 10, 2020 |
| Latest Amendment Date: | December 9, 2022 |
| Award Number: | 1954709 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Marian Bocea
mbocea@nsf.gov (703)292-2595 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 1, 2020 |
| End Date: | May 31, 2024 (Estimated) |
| Total Intended Award Amount: | $113,864.00 |
| Total Awarded Amount to Date: | $113,864.00 |
| Funds Obligated to Date: |
FY 2021 = $38,072.00 FY 2022 = $38,781.00 |
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
400 HARVEY MITCHELL PKY S STE 300 COLLEGE STATION TX US 77845-4375 (979)862-6777 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
3368 TAMU College Station TX US 77843-3368 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
01002122DB NSF RESEARCH & RELATED ACTIVIT 01002223DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Our world is essentially noncommutative in the sense that the order of actions often matters; for example, heating and cracking an egg can result in either a boiled egg or a fried egg, depending on the order of the operations. This is the reason why matrices, which encode noncommutativity in mathematics, are omnipresent in science. In many areas, such as control theory, quantum information theory and random matrix theory, the emerging questions about matrices and their ensembles are phrased so as to be independent of the matrix size. For example, a control system is designed as a black box, and its stability preferably does not depend on size of the input data (matrices) but only on the design and the structure of the system (a function of matrices). The common framework for such problems is provided by free analysis ("free" as in size-free), which studies functions in matrix variables. When such a function is built using only variables and arithmetic operations, it is called a noncommutative rational function. This project focuses on analytic, algebraic and geometric aspects of noncommutative rational functions and their evaluations on matrices. The goal is to apply novel synergistic techniques to answer fundamental open questions about noncommutative rational functions, apply their resolutions to semidefinite optimization and control theory, and accompany these theoretical results with efficient algorithms.
The aim of this project is twofold. On one hand, it considers questions about noncommutative rational functions that arise from free analysis and real algebraic geometry. Their common thread is the following: given a geometric feature of matrix evaluations of a noncommutative rational function, what can be deduced about its structure? This research focuses on positivity and singularity sets of noncommutative rational functions, their symmetries and existence of rational maps between them, with a view towards transforming non-convex (hard) problems in control theory and optimization into convex (easy) ones. Furthermore, this part of the project addresses natural extensions of noncommutative rational functions, such as noncommutative meromorphic functions and rational functions on operators acting on infinite dimensional spaces. On the other hand, noncommutative rational functions form a free skew field and are therefore related to several fundamental purely algebraic topics, such as the automorphism group of the free skew field, the free Lüroth problem, and the characteristic-free Freiheitssatz in a free algebra. The second part of this project proposes to apply ideas and techniques from free analysis to overcome these challenges.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Ensembles of matrices are ubiquitous models of operations in science. Their intrinsic features, independent of the reference frame and dimension, prominently appear in various parts of mathematics, such as control theory, quantum information theory, operator algebras and random matrix theory. This project investigated rational functions in several matrix or operator variables from this perspective, with the goal of understanding the structures governing their analytic and geometric behavior.
The PI and collaborators achieved major advances in understanding positivity of functions in matrix and operator variables. In particular, the PI resolved an analog of Hilbert's 17th problem for matrix-valued rational functions, showing that if a noncommutative rational function is positive semidefinite on its domain, then it is a sum of squares of noncommutative rational functions. Following this interplay of operator positivity and squares, the PI and its collaborators also established a comprehensive framework for noncommutative inequalities involving traces, ranging from algebraic certificates of positivity to algorithms for trace polynomial optimization.
The PI and collaborators also successfully applied function-theoretic ideas to resolve purely algebraic open problems on noncommutative rational functions. Firstly, the structure theory of invariant rational functions has been initiated; in particular, the PI and collaborators showed that noncommutative rational invariants for a finite solvable group are finitely generated (as opposed to noncommutative polynomial invariants, which are typical not). Secondly, the PI and collaborators constructed the universal skew field of fractions for tensor products of free algebras, answering an old question of P. M. Cohn.
Classification of ensembles of matrices up to similarity is an old and hard problem arising in representation theory, algebraic geometry and operator theory. The PI and collaborators showed that ranks of matrix-coefficient noncommutative polynomials form a family of invariants that completely separate matrix ensembles, and thus resolved a conjecture by D. Hadwin and D. Larson.
Finally, the above fundamental results also led to applications in quantum information theory. The PI and collaborators resolved certain nonlocality problems in quantum networks, devised a new algorithm for detecting entanglement in highly symmetric multipartite states, and proposed new schemes for device-independent certification of quantum measurements.
Last Modified: 07/02/2024
Modified by: Thomas Schlumprecht
Please report errors in award information by writing to: awardsearch@nsf.gov.
